Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$. Fix $\alpha \in \mathbb R$ and $\beta > 0$, and cobsider a derived function class on $H := \{\ell_f \mid f \in F\}$ on $X \times \{\pm 1\}$, where for each $f \in F$, the new function $\ell_f:X \times \{\pm 1\} \to \{0,1\}$ is defined by $$ \ell_f(x,y) = \begin{cases} 1,&\mbox{ if }|yf(x)-\alpha| \ge \beta,\\ 0,&\mbox{ otherwise.} \end{cases} $$

Question.What is the a good upper-bound on VC-dimension of $H$ in terms of some complexity measure associated with $F$ (e.g., Rademacher complexity of $F$, VC-dimension of $\mathrm{subgraph}(F) := \{A_f \mid f \in F\}$, where $A_f := \{x \in X \mid f(x) \le 0\}$, etc.) ?

I'm particularly interested in the case where $X=$ euclidean $\mathbb R^d$ (or unit-sphere in $\mathbb R^d$) and $F$ is the collection of functions $f:\mathbb R^d \to \mathbb R$ of the form $f(x) \equiv x^\top w + c$, for some $b \in \mathbb R$ and unit vector $w \in \mathbb R^d$.